In our dataset, empirical scaling exponents concord with the expected 3/4 optimum but predictions from underlying theory parameterized by branching ratios are smaller than expected

While our dataset is taxonomically diverse, certain species, particularly Prunus avium, Ulmus americana, and Quercus petraea are overrepresented in the dataset. Thankfully, these are also the highest quality trees in the dataset.

We can check briefly whether flattening the species distribution greatly affects our scaling results. Here, we reduce each species to n=5 with no observable changes in our main scaling results:

The success of the empirical estimate seems to be related to directly estimating volume scaling rather than using a ‘tip-counting’ approach. Here, we regress the total distal tip volume against the total volume for each subtree within a tree. This is in accord with the foundational assumptions of WBE theory regarding the scaling of metabolic service volumes with network size.

The question remains why theoretically computed exponents are so far below the optimum 3/4 level. We computed exponents using the following formula:

\(\theta = log(2^N) / (log(2^N) + log(1 - \xi^{N+1)}) - log((1-\xi)\xi^{N}))\) where \(\xi_{s} = 2\gamma\beta^2\) in the symmetric case and \(\xi_{a} = 2\gamma\beta^2 + 4\beta\Delta\beta\Delta\gamma + 2\gamma\Delta\beta^2\) in the asymmetric case

For the metabolic scaling exponent \(\theta\) to equal 0.75, symmetric traits occupy their own optima, corresponding to \(\beta \sim 0.7\) and \(\gamma \sim 0.79\). For all trees:

Averaged symmetrical traits for each tree reflect the observation that area preservation is broadly obeyed and space filling is broadly violated:

One important question is the role asymmetric traits are playing in these networks relative to classical symmetric traits.

Across trees, asymmetric traits cluster around 0 as expected, but are generally less important in determining scaling exponents. Most trees are ‘approximately symmetrical’ as seen by LiDAR scanners.

The plot below shows that symmetrical traits contribute most to ‘volume scaling’ \(\xi\) by a factor of five or six:

Despite the relatively low contributions of asymmetry to overall network scaling, we can also compare the relative contributions of positive and negative asymmetry across each network, potentially account for different mechanisms generating asymmetry in tree morphology:

Across all nodes in the dataset, the relative frequency of positive asymmetry is 17%, showing a large bias towards the presence of negative asymmetry in branching. However, for most trees in the dataset the relative contributions of positive and negative asymmetry are approximately equivalent, which may be contributing to ‘effective symmetry’ Interestingly, the positive/negative asymmetry signal is also reflected in symmetrical scaling traits, where nodes exhibiting negative asymmetry are contributing more to volume scaling, probably due to their greater relative frequency. Furthermore, this greater relative frequency appears to be driven by only a few trees.

Volume scaling \(\xi\) and the computed network depth \(N = log(n_{N})/log(2)\) are the two factors influencing scaling exponents computed from theory. Both factors are at play in determing our results:

Theoretical derivations assume that \(N \to \infty\) and that \(\xi \sim 0.8\). From these results, we conclude that smaller-than-expected symmetrical length scaling ratios \(\gamma\) are leading to lower-than-expected estimates of \(\theta\), in addition to the small network depth of real trees.

There is every indication that the formula \(N = log(n_{N})/log(2)\) could be underestimating N for asymmetric trees, in particular individuals experiencing disturbance and other forms of branch pruning.

However, no theory is available for reformulating the concept of branch order or branching ratio \(n\) on an asymmetric basis in the context of WBE, though the foundations of such theory might be available already (Horsfield 1974, 1976).

At this point what I would argue is that deviations in volume scaling reflect real biology / measurement error in LiDAR (variation in length scaling is real/unavoidable) and that deviations in network size are not real, or at least something we could account for in determining ‘effective’ metabolic scaling.

Here we implement a handful of algorithms for computing N from a network. Aside from the asymptotically symmetrical formula above, we implement a branch ordering scheme based on logarithmic binning of branches based on geometric characteristics (RADIUS or VOLUME), and algorithms for computing N at each node from the N of child branches. In particular: $N_{mean} = avg(N_{mean, 1} + N_{mean, 2}) $N_{min} = min(N_{min, 1} + N_{min, 2}) $N_{max} = max(N_{max, 1} + N_{max, 2})

$N_{asym} = ln(TIPS) / ln(N_{k}/k

The latter method is an asymmetric version of the above symmetrical computation, which simply recalculates the branching ratio based on the logarithmic binning and the methods of (Horsfield 1974, 1976)